THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS* John Lintner Introduction and Preview of Some Conclusions titive markets when utility functions are quad- HE effects of risk and uncertainty upon ratic or rates of return are multivariate normal.1 asset prices,upon rational decision rules We then note that the same conclusion follows for individuals and institutions to use in selecting from an earlier theorem of Roy's 19 without security portfolios,and upon the proper selection dependence on quadratic utilities or normality. of projects to include in corporate capital bud- The second section shows that if short sales are gets,have increasingly engaged the attention of permilted,the best portfolio-mix of risk assets professional economists and other students of the can be determined by the solution of a single capital markets and of business finance in recent simple set of simultaneous equations without years.The essential purpose of the present paper recourse to programming methods,and when is to push back the frontiers of our knowledge of covariances are zero,a still simpler ratio scheme the logical structure of these related issues,albeit gives the optimum,whether or not short sales under idealized conditions.The immediately are permitted.When covariances are not all following text describes the contents of the paper zero and short sales are excluded,a single quad- and summarizes some of the principal results. ratic programming solution is required,but The first two sections of this paper deal with sufficient. the problem of selecting optimal security port- Following these extensions of Tobin's classic folios by risk-averse investors who have the al- work,we concentrate on the set of risk assets ternative of investing in risk-free securities with held in risk averters'portfolios.In section III we a positive return(or borrowing at the same rate develop various significant equilibrium proper- of interest)and who can sell short if they wish. ties within the risk asset portfolio.In particular, The first gives alternative and hopefully more we establish conditions under which stocks will transparent proofs (under these more general be held long (short)in optimal portfolios even market conditions)for Tobin's important "sep-when "risk premiums"are negative (positive). aration theorem''that "..the proportion- We also develop expressions for different combi- ate composition of the non-cash assets is inde- nations of expected rate of return on a given pendent of their aggregate share of the invest- security,and its standard deviation,variance, ment balance..."(and hence of the optimal and /or covariances which will result in the same holding of cash)for risk averters in purely compe- relative holding of a stock,ceteris paribus.These "indifference functions"provide direct evidence "This paper is another in a series of interrelated theoretical and statistical studies of corporate financial and investment on the moot issue of the appropriate functional policies being made under grants from the Rockefeller Founda- relationships between"required rates of return" tion,and more recently the Ford Foundation,to the Harvard and relevant risk parameter(s)-and on the Business School.The generous support for this work is most gratefully acknowledged.The author is also much indebted related issue of how "risk classes"'of securities to his colleagues Professors Bishop,Christenson,Kahr,Raiffa, may best be delineated (if they are to be used).? and (especially)Schlaifer,for extensive discussion and com- Tobin [2,especially pp.82-85].Tobin assumed that mentary on an earlier draft of this paper;but responsibility for funds are to be a allocated only over "monetary assets"(risk- any errors or imperfections remains strictly his own. free cash and default-free bonds of uncertain resale price)and [Professor Sharpe's paper,"Capital Asset Prices:A Theory allowed no short sales or borrowing.See also footnote 24 be- of Market Equilibrium Under Conditions of Risk"(Journal of low.Other approaches are reviewed in Farrar [38). Finance,September 1964)appeared after this paper was in *It should be noted that the classic paper by Modigliani final form and on its way to the printers.My first section and Miller [16]was silent on these issues. Corporations were which parallels the first half of his paper(with corresponding assumed to be divided into homogeneous classes having the conclusions),sets the algebraic framework for sections II property that all shares of all corporations in any given class III and VI,(which have no counterpart in his paper)and for differed (at most)by a"scale factor,"and hence (a)were per- section IV on the equilibrium prices of risk assets,concerning fectly correlated with each other and(b)were perfect substi- which our results differ significantly for reasons which will be tutes for each other in perfect markets(p.266).No comment explored elsewhere.Sharpe does not take up the capital was made on the measure of risk or uncertainty (or other budgeting problem developed in section V below.] attributes)relevant to the identification of different "equiva- [13] This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
THE VALUATION OF RISK ASSETS AND THE SELECTION OF RISKY INVESTMENTS IN STOCK PORTFOLIOS AND CAPITAL BUDGETS * John Lintner Introduction and Preview of Some Conclusions T HE effects of risk and uncertainty upon asset prices, upon rational decision rules for individuals and institutions to use in selecting security portfolios, and upon the proper selection of projects to include in corporate capital bud- gets, have increasingly engaged the attention of professional economists and other students of the capital markets and of business finance in recent years. The essential purpose of the present paper is to push back the frontiers of our knowledge of the logical structure of these related issues, albeit under idealized conditions. The immediately following text describes the contents of the paper and summarizes some of the principal results. The first two sections of this paper deal with the problem of selecting optimal security port- folios by risk-averse investors who have the al- ternative of investing in risk-free securities with a positive return (or borrowing at the same rate of interest) and who can sell short if they wish. The first gives alternative and hopefully more transparent proofs (under these more general market conditions) for Tobin's important "sep- aration theorem" that ". . . the proportion- ate composition of the non-cash assets is inde- pendent of their aggregate share of the invest- ment balance . . " (and hence of the optimal holding of cash) for risk averters in purely compe- titive markets when utility functions are quad- ratic or rates of return are multivariate normal.' We then note that the same conclusion follows from an earlier theorem of Roy's 1191 without dependence on quadratic utilities or normality. The second section shows that if short sales are permitted, the best portfolio-mix of risk assets can be determined by the solution of a single simple set of simultaneous equations without recourse to programming methods, and when covariances are zero, a still simpler ratio scheme gives the optimum, whether or not short sales are permitted. When covariances are not all zero and short sales are excluded, a single quad- ratic programming solution is required, but sufficient. Following these extensions of Tobin's classic work, we concentrate on the set of risk assets held in risk. averters' portfolios. In section III we develop various significant equilibrium proper- ties within the risk asset portfolio. In particular, we establish conditions under which stocks will be held long (short) in optimal portfolios even when "risk premiums" are negative (positive). We also develop expressions for different combi- nations of expected rate of return on a given security, and its stand.ard deviation, variance, and/or covariances which will result in the same relative holding of a stock, ceteris paribus. These "indifference functions" provide direct evidence on the moot issue of the appropriate functional relationships between "required rates of return" and relevant risk parameter(s) - and on the related issue of how "risk classes" of securities may best be delineated (if they are to be used).2 *This paper is another in a series of interrelated theoretical and statistical studies of corporate financial and investment policies being made under grants from the Rockefeller Founda- tion, and more recently the Ford Foundation, to the Harvard Business School. The generous support for this work is most gratefully acknowledged. The author is also much indebted to his colleagues Professors Bishop, Christenson, Kahr, Raiffa, and (especially) Schlaifer, for extensive discussion and com- mentary on an earlier draft of this paper; but responsibility for any errors or imperfections remains strictly his own. [Professor Sharpe's paper, "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk" (Journal of Finance, September i964) appeared after this paper was in final form and on its way to the printers. My first section, which parallels the first half of his paper (with corresponding conclusions), sets the algebraic framework for sections II, III and VI, (which have no counterpart in his paper) and for section IV on the equilibrium prices of risk assets, concerning which our results differ significantly for reasons which will be explored elsewhere. Sharpe does not take up the capital budgeting problem developed in section V below.] 'Tobin [2I, especially pp. 82-85]. Tobin assumed that funds are to be a allocated only over "monetary assets" (risk- free cash and default-free bonds of uncertain resale price) and allowed no short sales or borrowing. See also footnote 24 be- low. Other approaches are reviewed in Farrar [38]. 2It should be noted that the classic paper by Modigliani and Miller [i6] was silent on these issues. Corporations were assumed to be divided into homogeneous classes having the property that all shares of all corporations in any given class differed (at most) by a "scale factor," and hence (a) were per- fectly correlated with each other and (b) were perfect substi- tutes for each other in perfect markets (p. 266). No comment was made on the measure of risk or uncertainty (or other attributes) relevant to the identification of different "equiva- [ 13 ] This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among uncertainty per se(as distinct from the effects of economists that relative risks are best measured diverse expectations),and to derive further by the standard deviation (or coefficient of implications of such uncertainty.In particular, variation)of the rate of return,*but in the simp-the aggregate market value of any company's lest cases considered-specifically when all equity is equal to the capitalization at the risk- covariances are considered to be invariant (or free interest rate of a uniquely defined certainty- zero)-the indifference functions are shown to equivalent of the probability distribution of the be linear between expected rates of return and aggregate dollar returns to all holders of its stock. their variance,not standard deviation.(With For each company,this certainty equivalent is variances fixed,the indifference function between the expected value of these uncertain returns less the ith expected rate of return and its pooled an adjustment term which is proportional to covariance with other stocks is hyperbolic.)their aggregate risk.The factor of proportion- There is no simple relation between the expected ality is the same for all companies in equilibirum rate of return required to maintain an investor's and may be regarded as a market price of dollar relative holding of a stock and its standard devia- risk.The relevant risk of each company's stock tion.Specifically,when covariances are non- is measured,moreover,not by the standard de- zero and variable,the indifference functions are viation of its dollar returns,but by the sum of the complex and non-linear even if it is assumed that variance of its own aggregate dollar returns and the correlations between rates of return on differ-their total covariance with those of all other stocks ent securities are invariant. The next section considers some of the impli- To this point we follow Tobin [21]and Marko- cations of these results for the normative aspects witz14]in assuming that current security prices of the capital budgeting decisions of a company are given,and that each investor acts on his own whose stock is traded in the market.For sim (perhaps unique)probability distribution over plicity,we impose further assumptions required rates of return given these market prices.In the to make capital budgeting decisions independent rest of the paper,we assume that investors'of decisions on how the budget is financed.The joint probability distributions pertain to dollar capital budgeting problem becomes a quadratic returns rather than rates of returns,and for programming problem analogous to that intro- simplicity we assume that all investors assign duced earlier for the individual investor.This identical sets of means,variances,and covari- capital budgeting-portfolio problem is formula- ances to the distribution of these dollar returns.ted,its solution is given and some of its more However unrealisic the latter assumption may important properties examined.Specifically, be,it enables us,in section IV,to derive a set of the minimum expected return (in dollars of ex- (stable)equilibrium market prices which at pected present value)required to justify the least fully and explicitly reflect the presence of allocation of funds to a given risky project is lent return"classes.Both Propositions I(market value of firm shown to be an increasing function of each of the independent of capital structure)and II(the linear relation following factors:(i)the risk-free rate of return; between the expected return on equity shares and the debt- (ii)the "market price of (dollar)risk";(iii)the equity ratio for firms within a given class)are derived from the above assumptions(and the further assumption that cor- variance in the project's own present value return; porate bonds are riskless securities);they involve no inter- (iv)the project's aggregate present value re- class comparisons,"..nor do they involve any assertion as turn-covariance with assets already held by the to what is an adequate compensation to investors for assuming company,and ()its total covariance with other a given degree of risk...."(p.279). This is,for instance,the presumption of Hirschleifer projects concurrently included in the capital [8,p.I13],although he was careful not to commit himself to budget.All five factors are involved explicitly this measure alone in a paper primarily focussed on other is- in the corresponding (derived)formula for the sues.For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk,see minimum acceptable expected rale of return on an Gordon [5,especially pp.69 and 761.See also Dorfman in investment project.In this model,all means [3,p.I29 fi.]and Baumol [2]. Except in dominantly "short"portfolios,the constant We also assume that common stock portfolios are not term will be larger,and the slope lower,the higher the (fixed) "inferior goods,"that the value of all other common stocks is level of covariances of the given stocks with other stocks. invariant,and any effect of changes in capital budgets on the sThe dollar return in the period is the sum of the cash covariances between the values of different companies'stocks is dividend and the increase in market price during the period. ignored. This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
14 THE REVIEW OF ECONOMICS AND STATISTICS There seems to be a general presumption among economists that relative risks are best measured by the standard deviation (or coefficient of variation) of the rate of return, but in the simp- lest cases considered - specifically when all covariances are considered to be invariant (or zero) - the indifference functions are shown to be linear between expected rates of return and their variance, not standard deviation.4 (With variances fixed, the indifference function between the ith expected rate of return and its pooled covariance with other stocks is hyperbolic.) There is no simple relation between the expected rate of return required to maintain an investor's relative holding of a stock and its standard devia- tion. Specifically, when covariances are non- zero and variable, the indifference functions are complex and non-linear even if it is assumed that the correlations between rates of return on differ- ent securities are invariant. To this point we follow Tobin [211 and Marko- witz [ 141 in assuming that current security prices are given, and that each investor acts on his own (perhaps unique) probability distribution over rates of return given these market prices. In the rest of the paper, we assume that investors' joint probability distributions pertain to dollar returns rather than rates of return5, and for simplicity we assume that all investors assign identical sets of means, variances, and covari- ances to the distribution of these dollar returns. However unrealisic the latter assumption may be, it enables us, in section IV, to derive a set of (stable) equilibrium market prices which at least fully and explicitly reflect the presence of uncertainty per se (as distinct from the effects of diverse expectations), and to derive further implications of such uncertainty. In particular, the aggregate market value of any company's equity is equal to the capitalization at the risk- free interest rate of a uniquely defined certainty- equivalent of the probability distribution of the aggregate dollar returns to all holders of its stock. For each company, this certainty equivalent is the expected value of these uncertain returns less an adjustment term which is proportional to their aggregate risk. The factor of proportion- ality is the same for all companies in equilibirum, and may be regarded as a market price of dollar risk. The relevant risk of each company's stock is measured, moreover, not by the standard de- viation of its dollar returns, but by the sum of the variance of its own aggregate dollar returns and their total covariance with those of all other stocks. The next section considers some of the impli- cations of these results for the normative aspects of the capital budgeting decisions of a company whose stock is traded in the market. For sim- plicity, we impose further assumptions required to make capital budgeting decisions independent of decisions on how the budget is financed.6 The capital budgeting problem becomes a quadratic programming problem analogous to that intro- duced earlier for the individual investor. This capital budgeting-portfolio problem is fornmula- ted, its solution is given and some of its more important properties examined. Specifically, the minimum expected return (in dollars of ex- pected present value) required to justify the allocation of funds to a given risky project is shown to be an increasing function of each of the following factors: (i) the risk-free rate of return; (ii) the "market price of (dollar) risk"; (iii) the variance in the project's own presentvalue return; (iv) the project's aggregate present value re- turn-covariance with assets already held by the company, and (v) its total covariance with other projects concurrently included in the capital budget. All five factors are involved explicitly in the corresponding (derived) formula for the minimum acceptable expected rate of return on an investment project. In this model, all means 6We also assume that common stock portfolios are not "inferior goods," that the value of all other common stocks is invariant, and any effect of changes in capital budgets on the covariances between the values of different companies' stocks is ignored. lent return" classes. Both Propositions I (market value of firm independent of capital structure) and II (the linear relation between the expected return on equity shares and the debt- equity ratio for firms within a given class) are derived from the above assumptions (and the further assumption that cor- porate bonds are riskless securities); they involve no inter- class comparisons, ". . . nor do they involve any assertion as to what is an adequate compensation to investors for assuming a given degree of risk. . . ." (p. 279). 3This is, for instance, the presumption of Hirschleifer [8, p. II 31, although he was careful not to commit himself to this measure alone in a paper primarily focussed on other is- sues. For an inductive argument in favor of the standard deviation of the rate of return as the best measure of risk, see Gordon [5, especially pp. 69 and 76I. See also Dorfman in [3, p. I29 ff.] and Baumol [2]. 4Except in dominantly "short" portfolios, the constant term will be larger, and the slope lower, the higher the (fixed) level of covariances of the given stocks with other stocks. 5The dollar return in the period is the sum of the cash dividend and the increase in market price during the period. This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
VALUATION OF RISK ASSETS 15 and (co)variances of present values must be cept in the final section,we assume that the calculated at the riskless rate r*.We also show interest rate paid on such loans is the same as he that there can be no "risk-discount"'rate to be used would have received had he invested in risk-free in computing present values to accept or reject savings accounts,and that there is no limit on the individual projects.In particular,the "cost of amount he can borrow at this rate.Finally (5) capital"'as defined (for uncertainty)anywhere he makes all purchases and sales of securities and in the literature is not the appropriale rale to use all deposits and loans at discrete points in time, in these decisions even if all new projects have the so that in selecting his portfolio at any "trans- same"“risk”as existing assets. action point,"each investor will consider only The final section of the paper briefly examines (i)the cash throw-off (typically interest pay- the complications introduced by institutional ments and dividends received)within the period limits on amounts which either individuals or to the next transaction point and(i)changes in corporations may borrow at given rates,by rising the market prices of stocks during this same costs of borrowed funds,and certain other "real period.The return on any common stock is de- world"complications.It is emphasized that fined to be the sum of the cash dividends received the results of this paper are not being presented plus the change in its market price.The return as directly applicable to practical decisions,be-on any portfolio is measured in exactly the same cause many of the factors which matter very way,includinginterest received or paid. siginificantly in practice have had to be ignored or assumed away.The function of these sim- Assumptions Regarding Investors plifying assumptions has been to permit a (1)Since we posit the existence of assets rigorous development of theoretical relationships yielding posilive risk-free returns,we assume that and theorems which reorient much current each investor has already decided the fraction of theory (especially on capital budgeting)and pro- his total capital he wishes to hold in cash and vide a basis for further work.?More detailed non-interest bearing deposits for reasons of conclusions will be found emphasized at numerous liquidity or transactions requirements.10 Hence- points in the text. forth,we will speak of an investor's capital as the stock of funds he has available for profitable I-Portfolio Selection for an Individual Investor: investment after optimal cash holdings have been The Separation Theorem deducted.We also assume that(2)each investor Market Assumptions will have assigned a joint probability distribution We assume that (1)eack individual investor incorporating his best judgments regarding the can invest any part of his capital in certain risk- returns on all individual stocks,or at least will have specified an expected value and variance to free assels (e.g.deposits in insured savings ac- countss)all of which pay interest at a common every return and a covariance or correlation to positive rate,exogeneously determined;and that every pair of returns.All expected values of (2)he can invest any fraction of his capital in any returns are finite,all variances are non-zero and or all of a given finite set of risky securities which finite,and all correlations of returns are less than one in absolute value (i.e.the covariance matrix are (3)traded in a single purely competitive markel,free of transactions costs and taxes,at is positive-definite).The investor computes the given market prices,?which consequently do not expected value and variance of the total return depend on his investments or transactions.We on any possible porlfolio,or mix of any specified also assume that (4)any investor may,if he amounts of any or all of the individual stocks,by wishes,borrow funds to invest in risk assets.Ex- forming the appropriately weighted average or 7The relation between the results of this paper and the sum of these components expected returns, models which were used in [I]and [I2]is indicated at the end variances and covariances. of section V. 1These latter decisions are independent of the decisions sGovernment bonds of appropriate maturity provide regarding the allocation of remaining funds between risk-free another important example when their "yield"is substituted assets with positive return and risky stocks,which are of for the word "interest." direct concern in this paper,because the risk-free assets with Solely for convenience,we shall usually refer to all these positive returns clearly dominate those with no return once investments as common stocks.although the analysis is of liquidity and transactions requirements are satisfied at the course quite general. margin. This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
VALUATION OF RISK ASSETS 15 and (co)variances of present values must be calculated at the riskless rate r*. We also show that there can be no "risk-discount" rate to be used in computing present values to accept or reject individual projects. In particular, the "cost of capital" as defined (for uncertainty) anywhere in the literature is not the appropriate rate to use in these decisions even iJ all new projects have the same "risk" as existing assets. The final section of the paper briefly examines the complications introduced by institutional limits on amounts which either individuals or corporations may borrow at given rates, by rising costs of borrowed funds, and certain other "real world" complications. It is emphasized that the results of this paper are not being presented as directly applicable to practical decisions, be- cause many of the factors which matter very siginificantly in practice have had to be ignored or assumed away. The function of these sim- plifying assumptions has been to permiit a rigorous development of theoretical relationships and theorems which reorient much current theory (especially on capital budgeting) and pro- vide a basis for further work.7 More detailed conclusions will be found emphasized at numerous points in the text. I -Portfolio Selection for an Individual Investor: The Separation Theorem Market Assumptions We assume that (1) each individual investor can invest any part of his capital in certain risk- free assets (e. g. deposits in insured savings ac- counts8) all of which pay interest at a common positive rate, exogeneously determined; and that (2) he can invest any fraction of his capital in any or all of a given finite set of risky securities which are (3) traded in a single purely competitive market, free of transactions costs and taxes, at given market prices,9 which consequently do not depend on his investments or transactions. We also assume that (4) any investor may, if he wishes, borrow funds to invest in risk assets. Ex- cept in the final section, we assume that the interest rate paid on such loans is the same as he would have received had he invested in risk-free savings accounts, and that there is no limit on the amount he can borrow at this rate. Finally (5) he makes all purchases and sales of securities and all deposits and loans at discrete points in time, so that in selecting his portfolio at any "trans- action point," each investor will consider only (i) the cash throw-off (typically interest pay- ments and dividends received) within the period to the next transaction point and (ii) changes in the market prices of stocks during this same period. The return on any common stock is de- fined to be the sum of the cash dividends received plus the change in its market price. The return on any portfolio is measured in exactly the same way, including interest received or paid. Assumptions Regarding Investors (1) Since we posit the existence of assets yielding positive risk-free returns, we assume that each investor has already decided the fraction of his total capital he wishes to hold in cash and non-interest bearing deposits for reasons of liquidity or transactions requirements.'0 Hence- forth, we will speak of an investor's capital as the stock of funds he has available for profitable investnment after optimal cash holdings have been deducted. We also assume that (2) each investor will have assigned a joint probability distribution incorporating his best judgments regarding the returns on all individual stocks, or at least will have specified an expected value and variance to every return and a covariance or correlation to every pair of returns. All expected values of returns are finite, all variances are non-zero and finite, and all correlations of returns are less than one in absolute value (i. e. the covariance matrix is positive-definite). The investor computes the expected value and variance of the total return on any possible portfolio, or mix of any specified amounts of any or all of the individual stocks, by forming the appropriately weighted average or sum of these components expected returns, variances and covariances. '0These latter decisions are independent of the decisions regarding the allocation of remaining funds between risk-free assets with positive return and risky stocks, which are of direct concern in this paper, because the risk-free assets with positive returns clearly dominate those with no return once liquidity and transactions requirements are satisfied at the margin. 7The relation between the results of this paper and the models which were used in [ii] and [I 2] is indicated at the end of section V. 8 Government bonds of appropriate maturity provide another important example when their "yield" is substituted for the word "interest." 9Solely for convenience, we shall usually refer to all these investments as common stocks, although the analysis is of course quite general. This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
16 THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for optimal mix of risk assets conditional on a given choices among different attainable combinations gross investment in this portfolio,and then for- of assets,we assume that(3)if any two mixtures mally proving the critical invariance property of assets have the same expected return,the inves-stated in the theorem.Tobin used more restric- tor will prefer the one having the smaller variance tive assumptions that we do regarding the avail- of return,and if any two mixtures of assets have able investment opportunities and he permitted the same variance of returns,he will prefer the no borrowing.u Under our somewhat broadened one having the greater expected value.Tobin [21,assumptions in these respects,the problem fits pp.75-76 has shown that such preferences are neatly into a traditional Fisher framework,with implied by maximization of the expected value different available combinations of expected of a von Neumann-Morgenstern utility function values and standard deviations of return on al- if either (a)the investor's utility function is con-ternative slock portfolios taking the place of cave and quadratic or (b)the investor's utility the original"production opportunity"set and function is concave,and he has assigned probabil- with the alternative investment choices being ity distributions such that the returns on all pos- concurrent rather than between time periods. sible portfolios differ at most by a location and scale Within this framework,alternative and more parameler,(which will be the case if the joint dis-transparent proofs of the separation theorem tribution of all individual stocks is multivariate are available which do not involve the actual normal). calculation of the best allocation in stocks over individual stock issues.As did Fisher,we shall Alternative Proofs of the Separation Theorem present a simple algebraic proofi2,set out the Since the interest rates on riskless savings logic of the argument leading to the theorem,and bank deposits ("loans to the bank")and on bor- depict the essential geometry of the problem.13 rowed funds are being assumed to be the same, As a preliminary step,we need to establish the we can treat borrowing as negative lending. relation between the investor's total investment Any portfolio can then be described in terms of in any arbitrary mixture or portfolio of individual (i)the gross amount invested in stocks,(ii)the stocks,his total net return from all his invest- fraction of this amount invested in each indivi- ments (including riskless assets and any borrow- dual stock,and (iii)the net amount invested in ing),and the risk parameters of his investment loans(a negative value showing that the investor position.Let the interest rate on riskless assets has borrowed rather than lent).But since the or borrowing be r*,and the uncertain relurn(divi- total net investment (the algebraic sum of stocks dends plus price appreciation)per dollar invested plus loans)is a given amount,the problem sim- in the given portfolio of stocks be r.Let w rep- ply requires finding the jointly optimal values resent the ralio of gross investment in stocks to for(1)the ratio of the gross investment in stocks uTobin considered the special case where cash with no return was the only riskless asset available.While he formally to the total net investment,and(2)the ratio of required that all assets be held in non-negative quantities the gross investment in each individual stock to (thereby ruling out short sales),and that the total value of risk the total gross investment in stocks.It turns out assets held not be greater than the investment balance available that although the solution of (1)depends upon without borrowing,these non-negativity and maximum value constraints were not introduced into his formal solution of the that of (2),in our context the latter is indepen- optimal investment mix,which in turn was used in proving the dent of the former.Specifically,the separalion invariance property stated in the theorem.Our proof of the theorem is independent of the programming constraints neglec- theorem asserts that: ted in Tobin's proof.Later in this section we show that when Given the assumptions about borrowing, short sales are properly and explicitly introduced into the set lending,and investor preferences stated earlier in of possible portfolios,the resulting equations for the optimum portfolio mix are identical to those derived by Tobin,but that this section,the optimal proportionate composition insistence on no short sales results in a somewhat more complex of the stock (risk-asset)portfolio (i.e.the solution programming problem (when covariances are non-zero),which to sub-problem 2 above)is independent of the may however,be readily handled with computer programs now available. ralio of the gross investment in stocks to the total net 12An alternative algebraic proof using utility functions inves!ment. explicitly is presented in the appendix,note I. Tobin proved this important separation theo- 1Lockwood Rainhard,Jr.hasalsoindependently developed and presented a similar proof of the theorem in an unpublished rem by deriving the detailed solution for the seminar paper. This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
16 THE REVIEW OF ECONOMICS AND STATISTICS With respect to an investor's criterion for choices among different attainable combinations of assets, we assume that (3) if any two mixtures of assets have the same expected return, the inves- tor will prefer the one having the smaller variance of return, and if any two mixtures of assets have the same variance of returns, he will prefer the one having the greater expected value. Tobin [21, pp. 75-761 has shown that such preferences are implied by maximization of the expected value of a von Neumann-Morgenstern utility function if either (a) the investor's utility function is con- cave and quadratic or (b) the investor's utility function is concave, and he has assigned probabil- ity distributions such that the returns on all pos- sible portfolios differ at most by a location and scale parameter, (which will be the case if the joint dis- tribution of all individual stocks is multivariate normal). Alternative Proofs of the Separation Theorem Since the interest rates on riskless savings bank deposits ("loans to the bank") and on bor- rowed funds are being assumed to be the same, we can treat borrowing as negative lending. Any portfolio can then be described in terms of (i) the gross amount invested in stocks, (ii) the fraction of this amount invested in each indivi- dual stock, and (iii) the net amount invested in loans (a negative value showing that the investor has borrowed rather than lent). But since the total net investment (the algebraic sum of stocks plus loans) is a given arnount, the problem sim- ply requires finding the jointly optimal values for (1) the ratio of the gross investment in stocks to the total net investment, and (2) the ratio of the gross investment in each individual stock to the total gross investment in stocks. It turns out that although the solution of (1) depends upon that of (2), in our context the latter is indepen- dent of the former. Specifically, the separation theorem asserts that: Given the assumptions about borrowing, lending, and investor preferences stated earlier in this section, the optimal proportionate composition of the stock (risk-asset) portfolio (i.e. the solution to sub-problem 2 above) is independent of the ratio of the gross investment in stocks to the total net investment. Tobin proved this important separation theo- ren by deriving the detailed solution for the optimal mix of risk assets conditional on a given gross investment in this portfolio, and then for- mally proving the critical invariance property stated in the theorem. Tobin used more restric- tive assumnptions that we do regarding the avail- able investment opportunities and he pernmitted no borrowing." Under our somewhat broadened assumptions in these respects, the problem fits neatly into a traditional Fisher framework, with different available combinations of expected values and standard deviations of return on al- ternative stock portfolios taking the place of the original "production opportunity" set and with. the alternative investment choices being concurrent rather than between time periods. Within this frarmework, alternative and more transparent proofs of the separation theorem are available which do not involve the actual calculation of the best allocation in stocks over individual stock issues. As did Fisher, we shall present a simple algebraic proof 12, set out the logic of the argument lea-ding to the theorem, and depict the essential geomretry of the problemr.13 As a preliminary step, we need to establish the relation between the investor's total investment in any arbitrary mixture or portfolio of individual stocks, his total net return from all his invest- nments (including risliless assets and any borrow- ing), and the risk parameters of his investment position. Let the interest rate on riskless assets or borrowing be r*, and the uncertain return (divi- dends plus price appreciation) per dollar invested in the given portfolio of stocks be r. Let w rep- resent the ratio of gross investment in stocks to "1Tobin considered the special case where cash with no return was the only riskless asset available. While he formally required that all assets be held in non-negative quantities (thereby ruling out short sales), and that the total value of risk assets held not be greater than the investment balance available without borrowing, these non-negativity and maximum value constraints were not introduced into his formal solution of the optimal investment mix, which in turn was used in proving the invariance property stated in the theorem. Our proof of the theorem is independent of the programming constraints neglec- ted in Tobin's proof. Later in this section we show that when short sales are properly and explicitly introduced into the set of possible portfolios, the resulting equations for the optimum portfolio mix are identical to those derived by Tobin, but that insistence on no short sales results in a somewhat more complex programming problem (when covariances are non-zero), which may however, be readily handled with computer programs now available. 12An alternative algebraic proof using utility functions explicitly is presented in the appendix, note I. 13 Lockwood Rainhard, Jr. has also independently developed and presented a similar proof of the theorem in an unpublished seminar paper. This content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
VALUATION OF RISK ASSETS 17 total net investment (stock plus riskless assets value of w).Since any expected return y can be minus borrowing).Then the investor's net obtained from any stock mix,an investor adher- return per dollar of total net investment will be ing to our choice criterion will minimize the (1)夕=(1-0)r*+7=r*十（行r*);0≤w<∞， variance of his over-all return a2,associated with where a value of w<1 indicates that the investor any expected return he may choose by confining holds some of his capital in riskless assets and all his investment in stocks to the mix with the receives interest amounting to (1-w)r*;while largest 0 value.This portfolio minimizes the w>1 indicates that the investor borrows to buy variance associated with any y (and hence any w stocks on margin and pays interest amounting to value)the investor may prefer,and consequently, the absolute value of (1-w)r*.From (1)we is independent of y and w.This establishes the determine the mean and variance of the net re- separation theorem1,once we note that our turn per dollar of total net investment to be: assumptions regarding available portfolios15 in- (2a）=r*+2w(rr*),and sure the existence of a maximum 0. It is equally apparent that after determining (2b)c3y=02g2. the optimal stock portfolio (mix)by maximizing Finally,after eliminating w between these two 0,the investor can complete his choice of an equations,we find that the direct relation be- over-all investment position by substituting tween the expected value of the investor's net the of this optimal mix in(3)and decide which return per dollar of his total net investment and over-all investment position by substituting the risk parameters of his investment position is:of the available(,)pairs he prefers by refer- (3a）y=r*+fa,where ring to his own utility function.Substitution (3b)0=(行-y)/a. of this best y value in (2a)determines a unique In terms of any arbitrarily selected stock port- best value of the ratio w of gross investment in folio,therefore,the investor's net expected rate the optimal stock portfolio to his total net of return on his total net investment is related investment,and hence,the optimal amount of linearly to the risk of return on his total net investments in riskless savings deposits or the investment as measured by the standard deviation optimal amount of borrowing as well. of his return.Given any selected stock portfolio, This separation theorem thus has four immedi- this linear function corresponds to Fisher's ate corrolaries which can be stated: "market opportunity line";its intercept is the risk-free rate r*and its slope is given by 0,which (i)Given the assumptions about borrowing is determined by the parameters and ar of the and lending stated above,any investor whose particular stock portfolio being considered.We choices maximize the expectation of any particu- also see from(2a)that,by a suitable choice of w, lar utility function consistent with these condi- the investor can use any stock mix (and its asso- tions will make identical decisions regarding the ciated "market opportunity line")to obtain an proportionate composition of his stock (risk-asset) expected return,as high as he likes;but that, portfolio.This is true regardless of the particular because of (26 )and (36),as he increases his in- ulility function!whose expectation he maximizes. vestment w in the (tentatively chosen)mix,the (ii)Under these assumptions,only a single standard deviation ox(and hence the variance point on the Markowitz "Efficient Frontier"is of the return on his total investment also relevant to the investor's decision regarding his becomes proportionately greater. investments in risk assets.17 (The next section Now consider all possible stock portfolios. See also the appendix,note I for a different form of proof. Those portfolios having the same0 value will 15 Specifically,that the amount invested in any stock in lie on the same "market opportunity line,"but any stock mix is infinitely divisible,that all expected returns on individual stocks are finite,that all variances are positive those having different 0 values will offer differ- and finite,and that the variance-covariance matrixispositive- ent"market opportunity lines"(between expected definite. return and risk)for the investor to use.The in- 16When probability assessments are multivariate normal, the utility function may be polynomial,exponential,etc. vestor's problem is to choose which stock port- Even in the "non-normal"case when utility functions are folio-mix (or market opportunity line or0 value) quadratic,they may vary in its parameters.See also the reference to Roy's work in the text below. to use and how intensively to use it(the proper 17When the above conditions hold (see also final para- This content downloaded from 202.120.21.61 on Mon,06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms
VALUATION OF RISK ASSETS 17 total net investment (stock plus riskless assets minus borrowing). Then the investor's net return per dollar of total net investment will be (1) y =(1 -w)r*+wf =r*+w(f-r*); O?<w< o, where a v alue of w < 1 indicates that the investor holds some of his capital in riskless assets and receives interest amounting to (1 -w)r*; while w> 1 indicates that the investor borrows to buy stocks on margin and pays interest amounting to the absolute value of (1-w)r*. From (1) we determine the mean and variance of the net re- turn per dollar of total net investment to be: (2a) y =r*+w(-r*), and (2b) a2y=W2.,2r. Finally, after eliminating w between these two equations, we find that the direct relation be- tween the expected value of the investor's net return per dollar of his total net investmnent and the risk parameters of his investment position is: (3a) y =r* +Ouy, where (3b) 0 = (r-S *) /0ru In terms of any arbitrarily selected stock port- folio, therefore, the investor's net expected rate of return on his total net investment is related linearly to the risk of return on his total net investment as measured by the standard deviation of his return. Given any selected stock portfolio, this linear function corresponds to Fisher's "market opportunity line"; its intercept is the risk-free rate r* and its slope is given by 0, which is determined by the parameters r and u,. of the particular stock portfolio being considered. We also see from (2a) that, by a suitable choice of w, the investor can use any stock mix (and its asso- ciated "market opportunity line") to obtain an expected return, y, as high as he likes; but that, because of (2b )and (3b), as he increases his in- vestment w in the (tentatively chosen) mix, the standard deviation oy (and hence the variance a2y) of the return on his total investment also becomes proportionately greater. Now consider all possible stock portfolios. Those portfolios having the same 0 value will lie on the same "market opportunity line," but those having different 0 values will ojjer dijTer- ent "market opportunity lines" (between expected return and risk) for the investor to use. The in- vestor's problem is to choose which stock port- folio-mix (or market opportunity line or o value) to use and how intensively to use it (the proper value of w). Since any expected return y can be obtained from any stock mix, an investor adher- ing to our choice criterion will minimize the variance of his over-all return o2y associated with any expected return he may choose by confining all his investment in stocks to the mix with the larges? 0 value. This portfolio minimizes the variance associated with any y (and hence any w value) the investor may prefer, and consequently, is independent of y and w. This establishes the separation theorem'4, once we note that our assumptions regarding available portfolios'5 in- sure the existence of a maximum 0. It is equally apparent that after determining the optimal stock portfolio (mix) by maximizing 0, the investor can complete his choice of an over-all investment position by substituting the 0 of this optimal mix in (3) and decide which over-all investment position by substituting of the available (y, ay) pairs he prefers by refer- ring to his own utility function. Substitution of this best y value in (2a) determines a unique best value of the ratio w of gross investment in the optimal stock portfolio to his total net investment, and hence, the optimal amount of investments in riskless savings deposits or the optimal amount of borrowing as well. This separation theorem thus has four immedi- ate corrolaries which can be stated: (i) Given the assumptions about borrowing and lending stated above, any investor whose choices maximize the expectation of any particu- lar utility function consistent with these condi- tions will make identical decisions regarding the proportionate composition of his stock (risk-asset) portfolio. This is true regardless of the particular utility functionl6 whose expectation he maximizes. (ii) Under these assumptions, only a single point on the Markowitz "Efficient Frontier" is relevant to the investor's decision regarding his investments in risk assets.17 (The next section 14See also the appendix, note I for a different form of proof. 15Specifically, that the amount invested in any stock in any stock mix is infinitely divisible, that all expected returns on individual stocks are finite, that all variances are positive and finite, and that the variance-covariance matrixispositive- definite. 16When probability assessments are multivariate normal, the utility function may be polynomial, exponential, etc. Even in the "non-normal" case when utility functions are quadratic, they may vary in its parameters. See also the reference to Roy's work in the text below. 17When the above conditions hold (see also final paraThis content downloaded from 202.120.21.61 on Mon, 06 Nov 2017 02:52:54 UTC All use subject to http://about.jstor.org/terms